Simplify the following expression: $y = \dfrac{5x^2+32x+35}{5x + 7}$
Answer: First use factoring by grouping to factor the expression in the numerator. This expression is in the form ${A}x^2 + {B}x + {C}$ First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(5)}{(35)} &=& 175 \\ {a} + {b} &=& &=& {32} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $175$ and add them together. The factors that add up to ${32}$ will be your ${a}$ and ${b}$ When ${a}$ is ${7}$ and ${b}$ is ${25}$ $ \begin{eqnarray} {ab} &=& ({7})({25}) &=& 175 \\ {a} + {b} &=& {7} + {25} &=& 32 \end{eqnarray} $ Next, rewrite the expression as $({A}x^2 + {a}x) + ({b}x + {C})$ $ ({5}x^2 +{7}x) + ({25}x +{35}) $ Factor out the common factors: $ x(5x + 7) + 5(5x + 7)$ Now factor out $(5x + 7)$ $ (5x + 7)(x + 5)$ The original expression can therefore be written: $ \dfrac{(5x + 7)(x + 5)}{5x + 7}$ We are dividing by $5x + 7$ , so $5x + 7 \neq 0$ Therefore, $x \neq -\frac{7}{5}$ This leaves us with $x + 5; x \neq -\frac{7}{5}$.